= Solution
Define the poloidal <Alfvén number> by
$$
\boxed{A^2=\frac{\mu_0\rho u_p^2}{B_p^2}
=\frac{\mu_0k^2}{\rho}},
$$
where $u_p=kB_p/\rho$. Thus, along a given field line,
$$
\boxed{\rho=\frac{\mu_0k^2}{A^2}};
$$
density decreases as the Alfvén number grows.
The azimuthal velocity representation gives
$$
u_\phi=r\omega+\frac{kB_\phi}{\rho}.
$$
Substitution into $\ell=ru_\phi-rB_\phi/(\mu_0k)$ yields
$$
\ell=r^2\omega+\frac{rB_\phi}{\mu_0k}(A^2-1).
$$
At the <Alfvén surface>, $A^2=1$. Smoothness requires the numerator of the solution for $B_\phi$ to vanish there, so
$$
\ell=\omega r_a^2.
$$
Solving away from the surface then gives
$$
\boxed{B_\phi
=\frac{\mu_0k\omega}{r}
\frac{r^2-r_a^2}{1-A^2}}.
$$
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